Nou, Alexandre Non-injectivity of the \(q\)-deformed von Neumann algebra. (English) Zbl 1060.46051 Math. Ann. 330, No. 1, 17-38 (2004). Summary: We prove that the von Neumann algebra generated by \(q\)-Gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a suitable vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called \(q\)-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non-injectivity under some assumption on the spectral set of the positive operator associated with the deformation. Cited in 21 Documents MSC: 46L65 Quantizations, deformations for selfadjoint operator algebras 46L54 Free probability and free operator algebras PDF BibTeX XML Cite \textit{A. Nou}, Math. Ann. 330, No. 1, 17--38 (2004; Zbl 1060.46051) Full Text: DOI arXiv